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ABSTRACT OF THE DISSERTATION

Semimartingales, Markov processes and their

Applications in Mathematical Finance

by Jin Wang

Dissertation Director: Paul Feehan

We show that the marginal distribution of a semimartingale can be matched by a

Markov process. This is an extension of Gyongys theorem to discontinuous semi-

martingale.

ii

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Acknowledgements

I would like to thank my advisor, Professor Paul Feehan. His support and kindness

helped me tremendously. I am deeply grateful for his constant encouragement and in-

sightful discussions in all these years at Rutgers.

I would like to thank all of my committee members, Professor Dan Ocone, Profes-

sor Richard Gundy and Professor Panagiota Daskalopoulos, for their comments and

encouragements.

I also want to thank the mathematics department of Rutgers for its support. Last

but not least, I want to thank all my friends at Rutgers, they made my PhD life colorful

and enjoyable.

iii

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Dedication

To my parents, Wang Xizhi, Wu Jinfeng.

iv

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Table of Contents

Abstract ii

Acknowledgements iii

Dedication iv

1. Introduction 1

2. Background 4

2.1. Volatility models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1. The Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2. Stochastic volatility models . . . . . . . . . . . . . . . . . . . . . 5

2.1.3. Dupires local volatility model . . . . . . . . . . . . . . . . . . . . 7

2.1.4. Volatility models with jumps . . . . . . . . . . . . . . . . . . . . 8

2.2. Mimicking theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1. Gyongys theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.2. Brunicks theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3. Bentata and Conts theorems . . . . . . . . . . . . . . . . . . . . 13

3. A Proof of Gyongys Theorem 18

3.1. Uniqueness of a solution to a partial differential equation . . . . . . . . 19

4. Forward Equation for a Semimartingale 24

4.1. Semimartingales and Markovian projection . . . . . . . . . . . . . . . . 24

4.2. Forward equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3. Construction of fundamental solutions . . . . . . . . . . . . . . . . . . . 34

4.4. Existence and uniqueness of weak solutions . . . . . . . . . . . . . . . . 47

4.5. Uniqueness of fundamental solutions . . . . . . . . . . . . . . . . . . . . 51

v

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5. Markov Processes and Pseudo-Differential Operators 53

5.1. Operator semigroups and Feller processes . . . . . . . . . . . . . . . . . 53

5.2. Pseudo-differential operators . . . . . . . . . . . . . . . . . . . . . . . . 58

5.3. Construction of a Markov process using a symbol . . . . . . . . . . . . . 61

6. Application of Brunicks Theorem to Barrier Options 66

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2. Application to up-and-out calls . . . . . . . . . . . . . . . . . . . . . . . 67

References 72

Vita 74

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Chapter 1

Introduction

In order to price and hedge nancial derivatives, stochastic process models of the dy-

namics of the underlying stocks have been introduced. The Black-Scholes model is

based on the assumption that the stock price process follows a geometric Brownian

motion with constant drift and volatility. It is well known that this model is too simple

to capture the risk-neutral dynamics of many price processes. Dupire [10] signicantly

improved the Black-Scholes model by replacing the constant volatility parameter by a

deterministic function of time and the stock price process. By construction, the asset

price process given by Dupires local volatility model has the same one-dimensional

marginal distributions as the market price process. Therefore, European-style vanilla

options whose values are determined by the marginal distributions can be priced cor-

rectly. In practice, the local volatility model is widely used to price not only vanilla

options, but also complex options with path dependent payoffs, even though such op-

tions cannot be priced correctly by Dupires model.

Dupires model is a practical implementation of a famous result of Gyongy [15].

Given an Ito process, Gyongy proved the existence of a Markov process with the same

one-dimensional marginal distributions as the given Ito process. The coefficients dened

in Gyongys process are given by Dupires local volatility model.

In [8] Brunick generalized Gyongys result under a weaker assumption so that the

prices of path dependent options can be determined exactly. For example, consider a

barrier option whose value depends on the joint distribution of the stock price and its

running maximum; Brunicks result shows that there exists a two-dimensional Markov

process with the same joint distributions. It gives a model process which is perfectly

calibrated against market data, but simpler than the market process.

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Bentata and Cont [3], [4] extended Gyongys theorem to semimartingales with

jumps. They showed that the ow of marginal distributions of a discontinuous semi-

martingale could be matched by the marginal distributions of a Markov process. The

Markov process was constructed as a solution to a martingale problem. However, their

proof of the main theorem is incomplete. Using this result, they derived a partial

integro-differential equation for call options. This is a generalization of Dupires local

volatility formula. We will discuss the details in Section 2.2.3.

Motivated by these results, we present a partial differential equation based proof to

the mimicking theorem of semimartingales in this thesis.

The thesis is organized as follows:

In chapter 2, we review some stochastic process models and mimicking theorems.

The rst section is dedicated to volatility models: the Black-Scholes model, stochastic

volatility models, including Heston model and SABR model, Dupires local volatility

model, and some jump models, including Mertons jump diffusion model and Bates

model. In the second section we introduce the theorems of Gyongy, Brunick, and

Bentata and Cont.

In chapter 3, we give a new proofs of Gyongys theorem. The proof is based on

a uniqueness result of a parabolic equation. We rst derive the partial differential

equation satised by the marginal distributions of an Ito process. Then we construct a

Markov process and show that the marginal distributions of this Markov process satisfy

the same partial differential equation. By the uniqueness result in [14], we conclude

that the Markov process has the same marginal distributions as the Ito process.

In chapter 4, we extend the partial differential equation proof of Gyongys theorem

for Ito process to the case of semimartingale processes. We rst derive the forward

equation satised by the probability density functions of a semimartingale and show

it is obeyed by the probability density functions of the mimicking process. This for-

ward equation is a partial integro-differential equation (PIDE). By analogy with the

construction of fundamental solutions to a partial differential equation, we construct

fundamental solutions of the PIDE using the parametrix method [14]. Through our

construction, we discover the conditions which ensure this equation has fundamental

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solutions. These conditions guarantee the existence of the transition probability density

function for the semimartingales. Then we derive and apply energy estimates to prove

uniqueness of the fundamental solution.

Pseudo-differential operators should be natural tools to study our partial integro-

differential equation. In chapter 5, we recall the relevant theory of pseudo-differential

operators, discuss their relationship with the martingale problem and indicate areas of

further research.

In the last chapter, we apply Brunicks result and derive a local volatility formula

for single barrier options. The exotic options market is most developed in the foreign

exchange market. This formula allows us to price exotic options in the FX market. And

these prices are consistent with the market prices of single barrier options we observe

in the market.

More ideas for further research

1. In Gyongys theorem, the drift and covariance processes are bounded, and the

covariance process satises the uniformly elliptic condition. However, the covari-

ance process in the Heston model is a CIR process which is neither bounded nor

bounded away from zero. So we plan to remove these constraints.

2. We plan to give a numerical illustration of Cont-Bentatas result using Monte

Carlo simulation.

3. We also plan to give a numerical illustration of Brunicks result by computation of

up-and-out call prices for(0)[0, ] using original and mimicking processes,where mimicking processes are geometric Brownian motion or Heston process.

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Chapter 2

Background

In this chapter, we review some models and theorems. The rst section is dedicated to

volatility models: the Black-Scholes model, stochastic volatility models, Dupires local

volatility model, and some jump models. In the second section we introduce Gyongys

theorem, Brunicks theorem, and Bentata and Conts result.

2.1 Volatility models

2.1.1 The Black-Scholes model

The Black-Scholes model [5] shows how to price options on a stock. An European call

option gives the right to its owner to buy at time one unit of the stock at the price ,

whereis called date of maturity and a positive number is called strike or exercise

price. The well known Black-Scholes formula gives the price of such an option as a

function of the stock price 0, the strike price , the maturity , the short rate of

interest , and the volatility of the stock . Only this last parameter is not directly

observable but it can be estimated from historical data. Call options are now actively

traded. The work of Black and Scholes on how to price call options seems irrelevant

as the prices are already known. However, this is not the case. First, there exist more

complex options, often called exotic options, a simple example that we will consider

later is a barrier option. These are not actively traded and therefore need to be priced.

More importantly the work of Black and Scholes shows that it is possible to create

a hedged position, consisting of a long position in the stock and a short position in

[calls on the same stock], whose value will not depend on the price of the stock. In

other words, they provide a method to hedge risk of a portfolio in the future. The

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replicating portfolios have to be rebalanced continuously in a precise way. The Black-

Scholes pricing is widely used in practice, because it is easy to calculate and provides

an explicit formula of all the variables.

However, prices obtained from the Black-Scholes formula are often differernt from

the market prices. In the model, the underlying asset is assumed to follow a geometric

Brownian motion with constant volatility . That is,

= + (),

where () is a Brownian motion. Given the market price of a call or put option, the

implied volatility is the unique volatility parameter to be put into the Black-Scholes

formula to give the same price as the market price. At a given maturity, options with

different strikes have different implied volatilities. When plotted against strikes, implied

volatilities exhibit a smile or a skew effect. Although volatility is not constant, results

from the Black-Scholes model are helpful in practice. The language of implied volatility

is a useful alternative to market prices. It gives a metric by which option prices can be

compared across different strikes, maturities, underlying, and observation times.

2.1.2 Stochastic volatility models

Stochastic volatility models are useful because they explain in a self-consistent way

why options with different strikes and expirations have different Black-Scholes implied

volatility. Under the assumption that the volatility of the underlying price is a stochastic

process rather than a constant, they have more realistic dynamics for the underlying.

Suppose that the underlying price and its variance satisfy the following SDEs,

= +

1,

= (, , ) + (, , )2,< 1, 2>=,

where is the instantaneous drift of, is the volatility of the volatility process and

is the correlation between random stock price returns and changes in , 1, 2 are

Brownian motions.

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The stochastic process followed by the stock price is equivalent to the one assumed

in the Black-Scholes model. As approaches 0, the stochastic volatility model becomes

a time-dependent volatility version of the Black-Scholes model. The stochastic process

followed by the variance can be very general. In the Black-Scholes model, there is

only one source of randomness, the stock price, which can be hedged by stocks. In the

stochastic volatility model, random changes in volatility also need to be hedged in order

to construct a risk free portfolio.

The most popular and commonly used stochastic volatility models are the Heston

model and the SABR model.

In the Heston model [16], the volatility follows a square root process, namely a CIR

process

= ( ) +

2, (2.1)

where is the mean long-term volatility, is the speed at which the volatility reverts

to its long-term mean and is the volatility of the volatility process.

the SABR model (stochastic alpha, beta, rho)[22] describes a single forward under

stochastic volatility :

= 1,

= 2,

The initial values0and 0are the current forward price and volatility, whereas1and

2 are two correlated Brownian motions with correlation coefficient . The constant

parameters, are such that 01,0.Once a particular stochastic volatility model is chosen, it must be calibrated against

existing market data. Calibration is the process of identifying the set of model param-

eters which most likely give the observed data. For instance, in the Heston model, the

set of model parameters,,, can be estimated from historic underlying prices.Once the calibration has been performed, re-calibration of the model is needed over

time.

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2.1.3 Dupires local volatility model

Given the computational complexity of stochastic volatility models and the difficulty

in calibrating parameters to the market prices of vanilla options, people want to nd a

simpler way to price exotic options consistently. Since before Breeden and Litzenberger

[7], it was well understood that the risk-neutral probability density function could be

derived from the market prices of European options. Dupire [10] and Derman and

Kani [9] show that under risk neutral measure, there exist a unique diffusion process

consistent with these distributions.

In Dupire s local volatility model, the constant volatility is replaced by a determin-

istic function of time and the stock price process, which is known as the local volatility

function. The underlying priceis assumed to follow a stochastic differential equation

= + (, ) (),

where is the instantaneous risk free rate and () is a Brownian motion.

The diffusion coefficients(, ) are consistent with the market prices for all option

prices on a given underlying. Dupires formula shows

2(, ; 0) = +

1

2

22

2

By construction, European style vanilla options whose values are determined by the

marginal distributions can be priced correctly. In practice, this model is used to de-

termine prices of exotic options which are consistent with observed prices of vanilla

options.

In [11], Dupire showed that the local variance is a conditional expectation of instan-

taneous variance. Assume the stochastic process for stock prices is

= + (),

he derived that

2(, ; 0) =[ =],

that is, local variance is the expectation of the instantaneous variance conditional on

the nal stock price being equal to the strike price . This equation implies that

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Dupires local volatility model is in fact a practical implementation of Gyongys theorem

which we will introduce in the next section.

2.1.4 Volatility models with jumps

Diffusion based volatility models cannot explain why the implied volatility skew is so

steep for very short expirations and why short-dated term structure of skew is incon-

sistent with any model. Therefore, jumps are necessary to be modeled. Examples of

such models are Mertons jump diffusion model [21] and Bates jump-diffusion stochas-

tic volatility model [2]. In these models, the dynamics of the underlying is easy to

understand and describe, since the distribution of jump sizes is known. They are easy

to simulate using Monte Carlo method.

Mertons jump diffusion model

Assume the stock price follow the SDE

= + + ( 1),

whereis a Poisson process, = 1 with probability , and = 0 with probability

1

. The jump size is lognormally distributed with mean log-jump and standard

deviation . We can rewrite the SDE as

= + + (+ 1),

with (0, 1). This allows us to get the probability density of

() =

=0

() exp ()22(2+2)!

2(2 + 2)

Prices of European options in this model can be obtained as a series where each terminvolves a Black-Scholes formula.

Bates model

Bates introduced the jump-diffusion stochastic volatility model by adding proportional

log-normal jumps to the Heston stochastic volatility model. The model has the following

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form

= +

1+

= ( ) +

2,

()1 and 2 are Brownian motions with correlation , and is a compound

Poisson process with intensity and log-normal distribution of jump sizes such that if

is its jump size then ln(1 + )(ln(1 +) 122, 2). The no-arbitrage conditionxes the drift of the risk neutral process, under the risk-neutral probability = .Applying Itos lemma, we obtain the equation for the log-price () = ln ,

() = ( 12

) +

()

1 + ,

where is a compound Poisson process with intensity and normal distribution of

jump sizes.

Bates model can also be viewed as a generalization of the Mertons jump diffusion

model allowing for stochastic volatility. Although the no arbitrage condition xes the

drift of the price process, the risk-neutral measure is not unique, because other pa-

rameters of the model, for example, intensity of jumps and parameters of jump size

distribution, can be changed.

2.2 Mimicking theorems

We want to construct simple processes which mimic certain features of the behavior of

more complicated processes.

Since the European option prices are uniquely determined by the marginal distri-

butions of the underlying price processes. In this section, we review some theorems in

which the marginal distributions of a general process are matched by a Markov process.

2.2.1 Gyongys theorem

Dupire derived the local volatility formula using the forward equation. In an earlier

work of Gyongy [15], he developed a result that is considered as a rigorous proof of the

existence of the local volatility model. He proved the following result.

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Theorem 2.1. (Gyongy [15, Theorem 4.6]) Let be an -dimensional Brownian

motion, and

() = + ()

be a-dimensional Ito process where is a bounded-dimensional adapted process, and

is a bounded -dimensional adapted process such that is uniformly positivedenite. There exist deterministic measurable function and such that

(, ()) = [()] a.s. for each,

(, ()) = [()] a.s. for each,

and there exists a weak solution to the stochastic differential equation:

= (, ) + (, )()

such thatL() = L(()) for all +, whereL denotes the law of a randomvariable and () denotes another Brownian motion, possibly on another space.

The assumptions on the drift and covariance processes in Gyongys theorem seem

too restrictive for many applications. For example, Atlan [1] computed the conditional

expectations for the Heston model using properties of the Bessel process and then used

Gyongys theorem to ensure the existence of a diffusion with the same marginal distri-

butions. However, the covariance process in the Heston model is a CIR process (2.1)

which is neither bounded nor bounded away from zero, so the conditions of Gyongys

theorem are not satised in this application.

Gyongys theorem is only valid for continuous Ito processes. It is important to

extend the result to processes with jumps.

2.2.2 Brunicks theorem

In practice, the local volatility model is also used to price complex options with path

dependent payoffs. However, the price cannot be uniquely determined by the marginal

distributions of the asset price process. For example, the price of a barrier option would

require knowledge of the joint probability distribution of the asset price process and its

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running maximum. Brunick [8] generalized Gyongys result under a weaker assumption

so that the prices of path dependent options could be determined exactly.

Denition 2.1. ([8, Denition 2.1])

Let : + (+, ) be a predictable process. We say that the path-functional is a measurably updatable statisticif there exists a measurable function

: + (+, )

such that (+ , ) = (, (, ); , (, )) for all (+, ), where the map : + (+, )(+, ) is dened by(, )() =(, ) ().

A measurably updatable statistic is a functional whose path-dependence can be

summed up by a single vector in . 1(, ) = () is an updatable statistic, as

1(+ , ) = 1(, ) + (, )(). Let () = sup (), we see that 2(, ) =

[(), ()] 2 is an updatable statistic as we can write

2( + , ) = [() + (, )(), max(), sup[0]

() + (, )()]

Theorem 2.2. ([8, Theorem 2.11])

Let be an-dimensional Brownian motion, and

() = + ()

be a-dimensional Ito process where is a left-continuous-dimensional adapted pro-

cess, and is a left-continuous -dimensional adapted process with

[

0

+ ] for all

Also suppose that : +

(+, )

is a measurably updatable statistic such that

the maps (, ) are continuous for each xed . Then there exist deterministicmeasurable function and such that

(, (, )) = [(, )] a.s. for Lebesgue-a.e. ,

(, (, )) = [(, )] a.s. for Lebesgue-a.e. ,

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and there exists a weak solution to the SDE:

= (, (, )) + (, (, ))()

such that L((, )) = L((, )) for all

+, whereL denotes the law of a

random variable and () denotes another Brownian motion.

Brunicks theorem is more general than Gyongys. First, the requirements on

and are weaker than the boundedness and uniform ellipticity in Gyogys theorem.

Secondly, this theorem implies the existence of a weak solution which preserves the

one-dimensional marginal distribution of path-dependent functionals.

Corollary 2.1. ([8, Corallary 2.16]) Let be an -dimensional Brownian motion,

and

() = + ()

be a-dimensional Ito process where is a left-continuous-dimensional adapted pro-

cess, and is a left-continuous -dimensional adapted process with

[

0

+ ] for all

Then there exist deterministic measurable function

and

such that

(, ()) = [()] a.s. for Lebesgue-a.e. ,

(, ()) = [()] a.s. for Lebesgue-a.e. ,

and there exists a weak solution to the SDE:

= (, ) + (, )()

such thatL() = L(())) for all +, whereL denotes the law of a random

variable and () denotes another Brownian motion.

This lemma relaxes the requirements on the coefficients of the Ito process in Gyongys

theorem.

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2.2.3 Bentata and Conts theorems

Bentata and Cont [3], [4] extended Gyongys theorem to a discontinuous semimartin-

gale. They showed that the ow of marginal distributions of a semimartingale could

be matched by the marginal distributions of a Markov process whose innitesimal

generator is expresses in terms of the local characteristics of. They gave a construc-

tion of the Markov process as the solution to a martingale problem for an integral-

differential operator. They applied these results to derive a partial integro-differential

equation for call options in a general semimartingale model. This generalizes Dupires

local volatility formula.

Consider a semimartingale given by the decomposition

() =0 + 0

+ 0

() + 0

1

() + 0

>1

(),

where is a -valued Brownian motion, is a positive, integer valued random

measure on [0, ) with compensator , = is the compensatedmeasure, and

are adapted processes in

and ().

Dene, for0,

(, ) =[ ( )

(

) =]

(, ) =[()() =]

( , ,) =[(, )() =]

Letbe an integer valued random measure on [0, ] with compensator( , , ()), = the associated compensated random measure, : [0, ) () is a measurable function such that (, )((, )) =(, ).

Theorem 2.3. ([3, Theorem 1])

If the function , and are continuous in (, ) on [0, ] , the stochastic

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differential equation

() =0+

0

(, ()) +

0

(, ())() +

0

1

()+

0 >1 (),

admits a weak solution(())[0] whose marginal distributions mimic those of:

() =() in distribution for any [0, ]

In the proof of this theorem, Bentata and Cont showed that, for any 2 function

with compact support

[(())] =(0) +

0

[(()) (, ())]

+1

2

0

[tr[2(())(, ())]]

+

0

[((() + ) (())

11(()))(,,())]

And it is easy to see that for the mimicking (), we also have

[(())] =(0) +

0

[(()) (, ())]

+1

2

0

[tr[2(())(, ())]]

+

0

[((() + ) (())

11(()))(,,())]

In order to show that() and() have the same marginal distributions, they asserted

that

[(())] =[(())],

for any 20 . However, they did not provide the proof of this statement in [3].

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In [3],is constructed as the solution ((())[0], 0) of the martingale problem

for an integro-differential operator

(, ) =

=1 (, )

(, ) +

=1(, )

2

2

2

(, )

+

[(, + ) (, ) =1

11

(, )]( , ,)

For any dom(),

=(()) () 0

(())

is a martingale.

The uniqueness if() is guaranteed by the following theorem.

Theorem 2.4. ([27]) If either

(i) for any [0, ], , , >0,

(ii) for any >0, >0, ( , ,) 1+

,

thenis the unique Markov process with innitesimal generator.

Bentata and Cont also derived a Dupire type formula for an asset price whose

dynamics under the pricing measure is a stochastic volatility model with random jumps,

=0

0

() +

0

() +

0

( 1)(),

where () is the discount rate, the spot volatility process and is a compensated

random measure with compensator

(; ) =(; , )

Theorem 2.5. ([4, Propostion 3])

Assume() and are locally bounded process and that for any > 0, there exists a

constant0 < 1

2(; , ),

(12)(; , ).Dene

(, ) =

[2 = ],

(,,) =[(, ) = ]

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The value(, ) at time of a call option with expiry > and strike > 0 is

given by

(, ) =[max( , 0)]

Then the call option price(, ) (, ) as a function of maturity and strike, isa solution (in the sense of distributions) of the partial integro differential equation on

[, ) (0, ),

=

2(, )2

2

2

2

+

( , , )

(, ) (, ) + (1 )

with initial condition(, ) = ( )+ for any >0.

This partial integro-differential equation generalizes the Dupire formula derived in

[10] for continuous process to the case of semimartingales with jumps. It implies that,

any arbitrage free option price across strike and maturity may be parameterized by a

local volatility function (, ) and a local Levy measure (,,).

They also gave some examples of stochastic models, including marked point pro-

cesses and time changed Levy processes.

In my thesis, we consider the same mimicking theorem of semimartingales as Bentata

and Cont did. However, my approach is totally different. In [3], Bentata and Cont

concluded that[(())] =[(())] for any2 function, therefore,() and()

have the same marginal distributions. In my proof, I show that (, ) and (, ),

the probability density functions of() and (), satises the same partial integro-

differential equation. And then I prove that this equation has a unique fundamental

solution under some assumptions for the diffusion and variance coefficients.

In [25], Ming Shi extended Gyogys theorem to pure jump processes and applied his

results to multi-name credit modeling. He also talked about the mimicking theorem

of semimartingales in Section 3.4 which was a joint work with me. We started with

the function () = for , . [(())] is the Fourier transform of themarginal distribution function (, ). We then used this fact to derive the forward

equation satised by (, ).

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In my thesis, I generalize this result by choosingas any2 function with compact

support.

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Chapter 3

A Proof of Gyongys Theorem

Gyogy proved his theorem by extending a result of Krylov [20]. We summarize his main

ideas of proof here. Consider the Green measure of an Ito process () with killing

rate (), which is dened by

() = +

0 1(())

0()

,for every Borel set , where 1 denotes the indicator function of the set , is a non-negative -adapted stochastic process. Denote the mimicking process by (),

Gyongy showed that the Green measure of (, ()) is identical the the Green measure

of (, ()). Let ()1, then we have

+0

(, ())

=

+0

(, ())

for every bounded non-negative Borel measurable function. Taking(, ) =()

with arbitrary non-negative constant and functions 0(), we get +0

[(())]=

+0

[(())],

this gives us

[(())] =[(())]

Hence it follows that the distribution of() and () are same for every . However,

the proof that() and() have the same Green measure is quite long and technical.

We present an intuitive proof of Gyongys theorem. The proof is based on the

uniqueness of solutions to a parabolic equation. Assume () has probability den-

sity functions (, ). We rst derive the partial differential equation satised by the

marginal distributions of an Ito process. Then we construct a Markov process and show

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that the marginal distributions of this Markov process satisfy the same partial differ-

ential equation. By uniqueness of solutions to a parabolic equation [14], we conclude

that the Markov process has the same marginal distributions as the Ito process.

3.1 Uniqueness of a solution to a partial differential equation

In this section, we give a proof of Gyongys theorem using uniqueness of solutions to a

partial differential equation. We shall use the following denition and theorems.

Denition 3.1. (Fundamental solutions of a parabolic equation)[14, Sections 1.1 &

1.6] Suppose , and are-valued functions on. A fundamental solution of

:=

(, ) 2

+

=1 (, ) + (, ) = 0 (3.1)in = [0, 1] is a -valued function (, ; , ) dened for all (, ) ,(, ), > , which satises the following conditions:(i) for xed(, ), it satises (3.1), as a function of(, ), , < < ;(ii) for every continuous-valued function()in such that [14, Section 1.6, Equa-

tion (6.1)],

() const. exp(2),

if, for some positive constant. Then

lim

(, ; , )()= ()

The integrals in Denition 3.1 exist only if 4( )< , where = 0 is denedas in [14, Section 1.2, Equation (2.2)] and the proof of Lemma 4.2 in this thesis. If

< 0

4(1 0)then the integrals in Denition 3.1 exist for all 0 < 1 [14, Section 1.6].

Denition 3.2. [14] A function(, ) is said to be Holder continuous with exponent

if

(, ) (0, 0) ( 0 + 02)

for any(, ), (0, 0)[0, 1] some constant >0.

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Theorem 3.1. [14, Theorem 1.15]

Assume that is parabolic and the coefficients of

,

, 2

, ,

,

are bounded, continuous -valued functions on [0, 1]; they satisfy a uniformHolder condition with exponent in . satises the ellipticity condition.

Then a fundamental solution of= 0 exists and

(, ; , ) = (, ; , ),

where is the formal adjoint of,

=

2

((, )) +

=1

((, )) + (, ) +

Theorem 3.2. ([13, Theorem 3.4]) The fundamental solution of (3.1) is unique.

Proof. Suppose that (, ; , ), (, ; , ) are two fundamental solutions for = 0.

Applying [14, Theorem 1.15], we see that

(, ; , ) = (, ; , ) =(, ; , )

for any (, ), (, )[0, 1] , > .And so (, ; , ) =(, ; , ) as desired.

Now we can proceed to give another proof of Gyongys theorem.

Theorem 3.3. Let be an -dimensional Brownian motion on a probability space

(,F, ), and

() = + ()

be an -valued Ito process, is a bounded -valued adapted process, and is a

bounded-valued adapted process such that is uniformly positive denite. Then

there exist deterministic measurable function(, ) and(, ) such that

(, ) = [() =] a.s. for0,

(, )(, ) = [() =] a.s. for,

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and

,

, 2

, ,

are bounded and Holder continuous with exponent , 0 < < 1. Assume () has

probability density functions(, ). Then there exists a weak solution to the stochastic

differential equation:

() =(, ()) + (, ())()

such that L(()) = L(()) for all > 0, where L denotes the law of a ran-

dom variable and ()denotes another Brownian motion, on another probability space

( , , ).

Proof. For any function 20 (), Ito formula shows

() () = 0

=1

()()

+1

2

=1

0

2

()[

, ]

=

0

=1

()( + ())

+

1

2

=1 0

2

()(()

())

Taking expectations on both side, we get

[()] =() +

0

=1

()

+

1

2

0

=1

2

()(()

())

,

=() +

0

=1

()+

1

2

0

=1

2

(())(()())

]

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Using iterated conditional expectation and conditioning on gives

[()] =() +

0

=1

()[]

+1

2

0

=1 2

(

)[(()())

]]

=() +

0

=1

()(, ) +

1

2

=1

2

()(, )

Denoting

() ==1

(, )

+

1

2

=1

(, ) 2

,

we have

[()] =() +

0

(())

(3.2)

If() has probability density functions (, ) , we can rewrite (3.2) as

()(, )= () +

0

()(, ) (3.3)

Taking derivatives of (3.3) with respect to , we get

()(, )=

()

(, )

=

=1(, )

(, )

+1

2

=1

(, ) 2

(, )

Integrating by parts on the right hand side gives

()(, ) (3.4)

=

=1

()

((, )(, )) +

1

2

=1

() 2

((, )(, ))

=

() =1

((, )(, )) +

1

2

=1

2

((, )(, ))

=

()((, ))

where is the formal adjoint of,

(, ) :==1

((, )(, )) +

1

2

=1

2

((, )(, ))

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Since (3.4) is true for any 20 function , then (, ) is a weak solution of

=(, )

Consider the process () dened as the solution to

() =(, ()) + (, ())()

Ito formula gives

[()] =() +

0

=1

()(, ) +

1

2

=1

2

()(, )

Let(, ) denotes the probability density functions of, we have

()(, )= () +

0 ()(, ) (3.5)Then we take derivatives with respect to and integrate by parts, we obtain

()(, )=

()((, ))

Therefore,(, ) is a solution to

=(, ),

in the weak sense, with initial condition (, 0) = (), where () is the Dirac delta

function.

We now consider an initial value problem

=(, ) (3.6)

(, 0) =()

Since (, ), (, ) are solutions of (3.6), it is enough to show that (3.6) has a

unique fundamental solution.

In [14, Section 1.4], a fundamental solution of

:=

,

is constructed by the parametrix method. By Theorem 3.2, we obtain the uniqueness

of the fundamental solution.

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Chapter 4

Forward Equation for a Semimartingale

In this chapter, we develop the main theorem of this dissertation. We rst introduce the

denition of semimartingales and dene the Markovian projection of semimartingales.

Then we derive the generator of a semimartingale and the backward equation. When the

semimartingale has a probability density function, we show that this function satises

the forward equation, the adjoint of the backward equation. Next we construct a

fundamental solution of the forward equation, through our construction, discover the

conditions which ensure that this equation has fundamental solutions. These conditions

guarantee the existence of the probability density functions for the semimartingale.

Finally, we show that the fundamental solution of the forward equation is unique.

Our existence and uniqueness result implies that the mimicking process has the same

marginal distributions as the original semimartingale.

4.1 Semimartingales and Markovian projection

In this section, we give the denition of a semimartingale and dene its Markovian

projection.

Denition 4.1. (Semimartingale, [24, Denition IV.15.1])

A processis a semimartingale on(, , (())0, )if it can be written in the form:

() =0+ + , (4.1)

where is a local martingale null at0 with cadlag paths and is an adapted process

with paths of nite variation, and the ltration is right-continuous.

An-valued process = (1, , ) is a semimartingale if each of its compo-

nents is a semimartingale.

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We emphasize that the decomposition (4.1) need not be unique. Semimartingales

are good integrators, forming the largest class of processes with respect to which the

Ito integral can be dened. The class of semimartingales is quite large, including, for

example, all Ito processes and Levy processes.

Proposition 4.1. (Ito decomposition for semimartingales, [4, Equation (2)])

On a ltered probability space (, , ()0, ), a semimartingale can be given by thedecomposition

() =(0)+

0

()+

0

()()+

0

1

()+ 0

>1

(),

(4.2)

where is a -valued Brownian motion, is a positive, integer valued random

measure on[0, ) with compensator,= is the compensated measure, has a density ( ,,), () and () are adapted processes valued in and

().

Proposition 4.2. (Ito formula for semimartingales, [23, Theorem 71])

Let (())0 be a semimartingale. For any twice continuously differentiable function

: ,

()

((0)) =

=1

0

((

))() +

1

2

=1

0

2

((

))[, ]

+

0

(() + ()) (())

=1

(())()

We want to construct a stochastic differential equation whose solution mimics the

marginal distributions of a semimartingale. We call the mimicking process aMarkovian

projection of a semimartingale.

Gyongy [15] showed that there exists a Markovian projection of an Ito process.

Brunick [8] generalized Gyongys result under a weaker assumption, and proved thatthere exists a Markovian projection of a two-dimensional process. Bentata and Cont [3],

[4] gave the existence of the Markovian projection of a semimartingale. The Markovian

projection is constructed as a solution to a martingale problem.

Here we present a partial differential equation based proof of mimicing theorem.

Consider a semimartingale () with decomposition (4.2) and further require

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(A1) ()() satises the uniformly ellipticity condition for any 0.That is, there is a constant >0 such that

=1(()())2,

for all and any 0 . Ellipticity thus means that the symmetricmatrix()() is positive denite, with smallest eigenvalue greater thanor equal to >0.

(A2) For any >0, () and () are bounded functions of[0, ].

Suppose the process () start at (0) = 0, and the density function of() is

(0, 0; , ), we denote the density by (0, 0; , ) and abbreviate by (, )

Theorem 4.1. (Markovian projection of a semimartingale)

Let() be a semimartingale starting with(0) =0, and() has a decomposition

() =(0)+

0

()+

0

()()+

0

1

()+ 0

>1

(),

where(), () satisfy (A1) and (A2). Assume() has probability density functions

(, ).

Dene

(, ) :=[()()() =]

(, ) :=[()() =]

(,,) :=[(,, )() =],

for all 0, , and any Borel set . Let be a positive, integer-valuedrandom measure on[0, ) with compensator, where has a density( ,,),and= is the associated compensated random measure.

We assume that

(i) , , and(,,) are continuous in(, ) on [0, ) ,

(ii) there is a constant, such that(, , )

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(iv) has a compact support.

Dene a process()

() =(0) +

0

(, ()) +

0

12 (, ())+

0 1 ()+

0

>1

()

where is an-valued Brownian motion. Then() is a Markovian projection ofthe semimartingale(), () and() have the same marginal distributions for any

0.

4.2 Forward equation

Suppose () is a semimartingale with generator , () is the process dened in

Theorem 4.1 with generator . We want to show that the density (0, 0; , ) of

() and (0, 0; , ) are solutions to the forward equation dened by .

Denition 4.2. The generator , 0 of a (time-inhomogeneous) process(())0is dened by

0() := lim0

[(())]

()

,

, 00,where is the expectation with respect to the probability law for () starting at

(0) = . The set of functions : such that the limit exists at (0, ) isdenoted byD0

(), whileD is the set of functions for which the limit exists for all

and00.

We assume has the property that D(, ) is dense, where (, ) isthe set of continuous functions from to which vanish at innity.

If() also obeys the conditions

(i) () is time-homogeneous;

(ii) (,D) satises the positive maximum principle;

(iii) ( ) is dense in (, ) for some >0;

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then by [18, Theorem 4.5.3], is a generator of a Feller semigroup and () is a

time-homogeneous Feller process.

The Courrege theorem [18, Theorem 4.5.21] shows that if: (, )(, )

is a linear operator satisfying the positive maximum principle. Then there exist func-

tions , , : and a kernel such that for 0 (, )

() =

=1

()2()

+

=1

()()

+ ()()

+

() ( )()

=1

()

( )( ) (,),

where(, ) with 01 and 1(0) = 1.The Courrege theorem gives us an example of the structure of . Conversely, when

has the structure described in the Courrege theorem, then is the generator of a

time-homogeneous Feller process.

Proposition 4.3. Letbe a semimartingale with decomposition (4.2) and satisfy the

assumptions in Theorem 4.1. If 20 (), that is is a2 function with compactsupport on, then D and the generator of() is

() ==1

(, )

+

1

2

=1

(, ) 2

+ ( + ) () 11

=1

( , ,)

Proof. Since is a semimartingale, with () =, the Ito formula gives

() ()

=

(()) () +12

=1

2

(())[, ]

+

0[(() + ()) (())

=1

(())()]

=

(())

() + ()() +1

() + >1

()

+1

2

=1

2

(())[, ]

+

0

(() + ()) (())

=1

(())()

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Simplifying, we obtain

() ()

=

(()) () +

(()) ()()

+1

2

=1

2

(())(()()) +

1

(())()+

>1

(())()

+

((() + ) (()) (()))()

We add the last two terms in the preceding equation and get

() ()=

(()) () +

(()) ()()

+1

2

=1

2

(())(()()) +

1

(())()+

((() + ) (()) 11 (()))()

Let denote[() =], when taking expectations involvingor [() =]when taking expectations involving . Taking expectations on both sides, we obtain

[()]

=() +

(()) ()

+

1

2

=1

2

(())(()())

+

((() + ) (()) 11 (()))(, )

We apply Fubini theorem and nd that

[()]

=() +

[(()) ()]

+1

2

=1

2

(())(()())

+

[

((() + ) (()) 11 (()))(, )]

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Using iterated expectations conditioned on (), we see that

[()]

=() +

[

((

))

[()

(

)]]

+1

2

=1

2

(())[(()()) ()]

+

[[

((() + ) (())

11 (()))(, )()]]

Simplifying gives

[()]

=() +

[(()) ((), )]

+1

2

=1

[ 2

(())((), )]

+

[

((() + ) (()) 11 (()))(,,())]

Therefore,

[()] (4.3)

=() +

=1

((), )

+1

2

=1

2

((), )

2

+

(() + ) (()) 11

=1

( , ,())

Similarly, suppose the process () starts with () =, then

[()]

=() +

=1

((), )

+1

2

=1

2

((), )

2

+

(() + ) (()) 11

=1

(,,())

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Then, by Denition 4.2, we get

() ==1

(, )

+

1

2

=1

(, ) 2

(4.4)

+ (( + ) () 11

=1

)(,,),

and

[(())] =() +

(())

(4.5)

And also, by equation (4.3), we have

[(())] =() +

(())

(4.6)

Since () =() with probability one, we have

lim

[(())() =] = (

)(),

and similarly for(),

lim

[(())() =] = (

)(),

since () =(

) with probability one.

Now we can derive the forward equation.

Proposition 4.4. (Forward equation)

Let be a semimartingale with decomposition (4.2) on with generator

() ==1

(, )

+

1

2

=1

(, ) 2

+ (( + ) () 11

=1

)(,,),

for20(), and assume that the probability measure of()has a density(, ),i.e.

[(())] =

()(, ), 20

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Then(, ) satises the forward equation

=

on (0, ),

where

is the formal adjoint of and is given by

() =1

2

=1

2

((, ))

=1

((, ))

+

(( , )(,, ) ()( , ,)

11=1

(( , ,)))

Proof. For 20 (), we have

[(())] =((0)) + 0

(())Since (, ) is the probability density function of(), we can rewrite this equation

as

()(, )= ((0)) +

0

()(, )

Taking derivatives with respect to of both sides of the preceding equation, we get

()(, )

=

()

(, )

=

=1

(, )

(, ) +

1

2

=1

(, ) 2

(, )

+ (( + ) () 11=1

)( , ,)(, )

Integrating by parts on the right hand side, we get

()(, )

=

=1

()

((, )(, )) +

1

2

=1

() 2

((, )(, ))

+

( + ) () 11

=1

(,,)(, )

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Consider the last integral in the preceding equation. For the rst term, we need to

shiftby, and for the last term, we integrate by parts once,

( + ) () 11

=1

( , ,)(, )

=

(()(,, )( , ) ()( , ,)(, )

+ ()11

=1

(( , ,)(, )))

Thus,

()(, )

= ()

=1

((, )(, )) +

1

2

=1

2

((, )(, )) +

(( , , )( , ) (,,)(, )

+ 11

=1

(( , ,)(, )))

=

()()(, )

We obtain

()

(, )=

()(, )=

() (, ),

which we can write it in terms of the 2 inner product,

,

=, =, ,

where

is the formal adjoint of and (, ) is a solution of

= (, )

in the weak sense as in [12, Section 7.2].

Similarly, (, ), the density function of the process () also satises the equation

= (, )

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4.3 Construction of fundamental solutions

In this section, we will consider the fundamental solutions of the forward equation

dened by the generator .

The construction of fundamental solutions of parabolic differential equations is de-

scribed by Friedman in [14, Sections 1.2 & 1.4] in the case of bounded domains

and extended to unbounded domains (including ) in [14, Section 1.6]. Wefollow his approach and construct fundamental solutions of the forward equation we

derived in the previous section, noting that the forward equation is a partial integro-

differential equation. Construction of fundamental solutions of integro-differential op-

erators of this kind have also been described by Garroni and Menaldi [?, Theorem

4.3.6] for bounded domains in and extended to the case of unbounded domains (in

particular,) by [?,4.3.3].Dene by the expression

:=

we simply this expression before we proceed:

(, ) =(, ) +

=1((, )(, ))1

2

=1((, )(, ))

( , )( , , ) (, )( , ,)

+ 11

=1

(( , ,)(, ))

This leads to

(, ) =(, ) 12

=1

(, ) (, )

+=1

((, ) 12

=1

(, ) )(, ) +=1

((, )12

=1

(, ))(, )

( , )( , , )+(, )

(,,)

(, )

11

=1

( , ,) (, )

11

=1

(,,)

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Denote

(, ) :=1

2(, ),

(, ) :=(, ) 12

=1 (, ) 11

=1 ( , ,),(, ) :=

=1

((, )1

2

=1

(, ))

+

(,,) 11=1

( , ,)

Then,

(, ) =(, )

=1

(, ) (, ) +=1

(, )(, )

+ (, )(, )

(, )(, , ),

Consider the equation

=0, (4.7)

where the coefficients , , and are dened on [0, ]. The domain can be unbounded and includes the important case = . Throughout thissection we assume:

(A3) is parabolic in .

(A4) The coefficients of are continuous functions in and, in addition, for all

(, ), (0, 0), there exist constants 0 < , 0<

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Denition 4.3. (Fundamental solutions of a forward equation)

A fundamental solution of a forward equation= 0 in is a function(, ; , )

dened for all(, ), (, ), > , which satises the following conditions:(i) For xed (, )

, it satises, as a function of (, ),

, < < , the

equation= 0;

(ii) For every continuous function in obeying the growth condition in Denition

3.1, if then

lim

(, ; , )()= ()

For a Ito process, the operator is

=

=1

(, ) +

=1

(, ) + (, ),

for a semimartingale, the operator is

=

=1

(, ) +=1

(, )+ (, )

(, )(, , )

We adapt the parametrix method in [14, Section 1.2] for the construction of funda-

mental solutions of linear second-order parabolic PDEs to the construction of funda-

mental solutions of our linear second-order parabolic PIDE; see also [?, Chapter V] and

[?, Section IV.11]. Existence of fundamental solutions for parabolic PIDEs of the kind

examined in this thesis is also proved in [?], using related methods. The construction of

fundamental solutions for our parabolic PIDE closely mirrors that of parabolic PDEs

due to the assumption on the density dening the integral term.

First we introduce the function (, ; , ), for > ,

(, ; , ) =(, ) 1

( )

2

exp 14( )

=1 (, )( )( ) ,where

(, ) = 1

(2

)[det((, ))]

12 ,

and(, ) is the inverse matrix to (, ). The function is called theparametrix.

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For each xed (, ), the function (, ; , ) satises the following equation with

constant coefficients,

0(, ) :=

=1(, ) = 0,

and also satises the following proposition.

Proposition 4.5. ([14, Section 1.2 Theorem 1])

Let be a continuous function in obeying the growth condition in Denition 3.1.

Then

(,,) :=

(, ; , )(, )

is continuous function in(,,), , 0 < and

lim

(,,) =(, ),

uniformly with respect to (, ), , 0< , where is any closed subset of.

From now on, we use to denote the fundamental solutions of the forward equation,

because is traditional for transitional probability density function. It follows from

Proposition 4.5 that property (ii) in Denition 4.3 of the fundamental solution is also

satised for= .

In order to construct a fundamental solution for = 0, we view 0 as a rst

approximation to and we view as the principal part of the fundamental solution

. We then try to nd in the form

(, ; , ) =(, ; , ) +

(, ; , )(, ; , ),

where is to be determined by the condition that satises the equation ,

0 =(, ; , )

=(, ; , ) +

(, ; , )(, ; , )

We rst consider the term (, ; , ). For xed (, ), 0(, ; , ) = 0, and

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so

(, ; , ) =(, ; , ) 0(, ; , ) (4.8)

=

=1 (, ) +

=1 (, ) + (, ) (4.9)

(, ; , )(, , ) +

=1

(, ) (4.10)

=

=1

((, ) (, )) +=1

(, )+ (, )

(4.11)

(, ; , )(, , ) (4.12)

We need the following lemma to calculate

(, ; , )(, ; , )

Lemma 4.1. ([14, Section 1.3 Lemma 1 ])

Letbe a continuous function inobeying the growth condition in Denition 3.1 and

locally Holder continuous in, uniformly with respect to , and

(, ) =

0 (, ; , )(, )Then

(, ) =

0

(, ; , )(, ),

2

(, ) =

0

2

(, ; , )(, ),

(, ) =(, ) +

0

=1

(, ) 2

(, ; , )(, )

If is such that Lemma 4.1 applies to (, ) := (, ; , ), then

(, ) =

(, ; , )(, ; , )

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Consequently,

(, ) =

=1

(, ) +=1

(, )+ (, )

(, )(, , )= (, ; , ) +

0

=1

(, ) 2

(, ; , )(, ; , )

+

=1

(, ) +=1

(, ) + (, )

(, ; , )(, ; , )(, , )

Therefore,

(, ) = (, ; , ) + [

=1(, ) (, )] (4.13)

+=1

(, )+ (, ) (4.14)

(, ; , )(, ; , )(, , )

(4.15)

= (, ; , ) +

(, ; , )(, ; , ) (4.16)

Combining (4.8) and (4.13) together, we get

0 =(, ; , )

=(, ; , ) + (, )

=(, ; , ) + (, ; , ) +

(, ; , )(, ; , )

Therefore,

(, ; , ) =(, ; , ) +

(, ; , )(, ; , ) (4.17)

Thus, for each xed (, ), the function (, ; , ) is a solution of a Volterra integral

equationwith singular kernel (, ; , ).

Before we proceed to prove the existence of the fundamental solution, we give two

useful lemmas. Lemma 4.2 is modeled after [14, Inequality (4.3), Section 1.4], but we

need to evaluate an extra term, namely the integral term

(, ; , )(, , )

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in the PIDE.

Lemma 4.2. Let the notation be as above. Then,

(, ; , )

const

( )

1

+22

(4.18)

where, are constants, 1 2 < < 1.

Proof. From the denition of, we have

(, ; , ) =

=1

((, ) (, )) +=1

(, )+ (, )

(, ; , )(, , ),

where

(, ; , ) = (, ) 1

( )2 exp 1

4( )

=1

(, )( )( ) ,

and

(, ) = 1

(2

)[det((, ))]

12

Here () is the inverse matrix of () and satises the ellipticity condition, that is,

there exists a constant >0 such that

=1

( )( ) 2

For xed (, ), if 0< < 2 , then

(, ; , ) const( )2 exp

2

4( )

(4.19)

= const

( )1

2 2( )2 exp

2

4( )

const

( )

1

2

The last inequality is true because

2( )2 exp

2

4( )

=

2exp

4(

)2

is bounded as a function of (, ; , ) in , where =(, ; , ) : = , = as long as 0< < 2 for any 1.

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Let:= + (1 2 ), then 1 2 < 0.We assume that (, ) is bounded as a function of (, ), thus

(, )(, ; , ) const( )

1

+22 (4.20)

Similarly, we can apply this trick to and and get

(, ; , ) const

( )2+1 exp

2

4( )

,

const( ) 1 +22 ,

2

(, ; , )const

2

( )2+2 exp

2

4( )

,

const( )

1

+22

By assumption (4), (, ) is bounded and(, ) (, ) const , weget

(, )

(, ; , ) const( )

1 +22 , (4.21)

((, ) (, )) 2

(, ; , ) const

( )1

+22 (4.22)

Now we consider the integral

(, ; , )(, , )

Because is compactly supported as a function of (, )

, for

21

we have

(, ; , )(, , )const

(, ; , )

const

1

( )2 exp[ 24( )]

const,

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where the last estimate follows because the function

1

( )2 exp

2

4( )

is integrable.

When > 21, by changing the variables and , we get

(, ; , )(, , )=

( , ; , )( , , )

Since is bounded function with compact support, we have

(, ; , )(, , )const 11

( , ; , )

By equation (4.19), we get

11 ( , ; , )const 1

1

1

( )2 exp[

2

4( ) ]Because > 21 and< 1, so

1 12 = 1

2 ,

2 14 2,

thus

1

1

1

( )2

exp[ 2

4(

) ]

1

1

1

( )2

exp[4 24(

)

]

Using the same trick applied in equation (4.19), we get

1

( )2 exp[2 24( ) ]const

1

( )2 exp[2 24( ) ]

Therefore,

(, ; , )(, , )

also satises the inequality

(, ; , )(, , ) const( ) 1 +22 , (4.23)Adding inequalities (4.20), (4.21), (4.22), (4.23) together, we obtain

(, ; , ) const( )

1

+22 ,

when 1 2 < < 1.

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The following Lemma is based on [14, Lemma 2, Section 1.4] which only holds for

bounded domain. We extend it to the case of an unbounded domain.

Lemma 4.3. Suppose that is a domain in , which could be , 0 < < ,

0< < , and + > then for any, , =, we have

1

Proof. Let1=

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Theorem 4.2. There exists a solution of equation (4.17) of the form

(, ; , ) ==1

()(, ; , ), (4.24)

where

()1 := ,

()+1(, ; , ) :=

[(, ; , )]()(, ; , )

Proof. Using Lemma 4.2 and Lemma 4.3, we get 1

()2(, ; , )=

[(, ; , )]()(, ; , )

const

1

(

)

1

+22

1

(

)

1

+22

const

1

( )1

( )

1

+221

+22

const 1( )21

1

+2(22) ,

when 2 >1 and +2(2 2 )> 0. Since 1 2 < 2 and + 3(2 2 )> 0.We know after nite steps, we arrive at some 0 for which 0 < 0 1, and

+ 0(2 2 )< 0, thus

()0(, ; , ) const

From0, we proceed to prove by induction on , assume that

()+0(, ; , ) 0[( )1]

(1 + (1 )) ,

1See Remark 4.1 for additional details for the case of unbounded domains

.

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where 0, are constants and () is the gamma function. For = 0, this follows

from()0(, ; , ) const. Assuming now that it holds for some integer 0,and using Lemma 4.2 we get

()+1+0(, ; , ) const.0

((1 ) + 1) ( )( )(1)Substituting = into the preceding expression and using the formula 1

0(1 )11= ()()

( + ),

we obtain

( )( )(1)= 10

( )1( + 1)1(1 )

= ( )1( + 1) (1 )(1 + (1 ))(1 + (1 )( + 1))

Thus

()+1+0(, ; , ) 0[( )1]+1

(1 + (1 )( + 1)) ,

and the induction holds for + 1.

It follows that

(, ; , ) =

=1

(

)(, ; , )

=

0=1

()(, ; , ) +

=1

()0+(, ; , )

const( )

1

+22 ,

and the series converges.

From Theorem 4.2, it follows that the series expansion (4.24) for (, ; , ) con-

verges and that integral term in (4.17) is equal to=1

(, ; , ) ()(, ; , )

Therefore,

(, ; , ) =(, ; , ) +

(, ; , )(, ; , ),

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satises (4.7), so property (i) in Denition 4.3 holds. By Proposition 4.5, property

(ii) also holds. Therefore(, ; , ) is a fundamental solution of (4.7). Compare the

statements and proofs of [14, Section 1.4, Theorem 8] for bounded domains and [14,

Section 1.6, Theorem 10] for unbounded domains.

Remark 4.1. Friedman notes in [14, Section 1.6] that the construction of the fun-

damental solution, (, ; , ), extends from the case of a bounded domain

to an unbounded domain and in particular = . We briey summarizeone approach to the changes for unbounded domains here and refer the reader to stan-

dard references for further details [?, Chapter V] and [?, Section IV.11]. The estimate

in Lemma 4.2 is replaced [?, Chapter V, Equation (3.19)], [?, Chapter 4, Equation

(11.17)] by the better behaved

(, ; , ) ( ) 12 (2) exp

2

,

where (0, 1) is the Holder constant, as before, and , are positive constants;Lemma 4.3 will not be used. This estimate is standard when no integral term appears

in the denition of , while the proof of Lemma 4.2 shows that the integral term in

(, ; , ) also obeys this estimate; indeed, the proof of Lemma 4.2 yields

(, ; , )(, , )( )2 exp 2

( ) 12 (2) exp

2

,

using 12( 2)(1, 12) and0 , and this bound replaces (4.23).Next, the estimate appearing in the proof of Theorem 4.2 for the term()(, ; , )

in the innite series dening (, ; , ), obtained by the iterative method of solving

the Volterra integral equation (4.17), is replaced by [?, Chapter V, Equation (3.22)], [?,

Chapter 4, Equation (11.25)]

()(, ; , )

(1)2 (2)

(2)( ) 12 (2) exp

2

,

for1, where()1:= . These estimates for()(, ; , ) ensure uniformconvergence of the series in the statement of Theorem 4.2 for >0 and yields the

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estimate

(, ; , ) ( ) 12 (2) exp

2

,

just as in [?, Chapter 4, Equation (11.26)].

4.4 Existence and uniqueness of weak solutions

In this section, we will show the existence and uniqueness of weak solutions of the

partial integral equation (4.7)

= 0

Letbe an open subset of, and set = (0, ]. can be unbounded in, and the special case = is of particular importance.

Let

=

=1

((, )) +=1

((, ))+ (, ) +

(, )(, , ),

where(,,) : [0, ] .We will study the following parabolic equation with initial and boundary conditions

+ = in (4.25)

=0 on [0, ]= on = 0

We assume that the coefficients of satisfy the following conditions (A5):

=1

(, )2 for all (, ), (4.26)

, , () (4.27)

(0, ; 2(

)) (4.28)

2() (4.29)

2() (4.30)

Remark 4.2. (0, ; 2( )) means

(, , )2() :=

2(,,)

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is bounded by a nite constant for a.e. [0, ].

To apply a theorem in [26, Section 3.2], we need the following lemma.

Lemma 4.4. Suppose , , satisfy (A5). Then there exist positive constants and

depending only on the coefficients of such that

(,) + 2()10 ()

for a.e. [0, ] and all1().

Proof. Since satises the elipticity condition (4.27), we have

(, )22(), (4.31)

for a.e. [0, ] and some positive constant .

Since(), then for a.e [0, ] the Cauchy inequality yields

(, )(, )()2()1()

Then for any 0, there exists > 0 such that

(, )(, )() 2()2(1()) (4.32)

22() 21() (4.33)

Because(), then for a.e [0, ], there exists a constant 1 such that

2(, )122() (4.34)

Since

2(,,)

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is bounded by a nite number for a.e. [0, ], the Cauchy inequality gives

()()(, , ) (4.35)

(2())

12 ((()(, , ))

2)12

2()(

(

2()

2(, , ))) 12

=22()

2(, , )

222()

for a.e [0, ] and some constant 2.By combining (4.31), (4.32), (4.34) and (4.35), we obtain

< , >2() = (, ) + (, )(, )() + 2

+

()()(, , )

22() 22() 21() 12() 222()

Setting := 2, := + 1+ 2, we obtain

< , >2() +2()10 (),

where :=2 This completes the proof.

Let be a separable Hilbert space with dual ; then2(0, ; ) is a Hilbert space

with dual 2(0, ; ). Assume that for each [0, ] we are given a continuousbilinear form (; , ) on or, equivalently, an operator() (, ),

()() =(; , ), , , [0, ],

such that for each pair , the function (; , ) is in (0, ; ). Assume is

a Hilbert space identied with its dual and that the embedding is dense andcontinuous;hence by restriction. Finally, let 2(0, ; ) and 0 begiven.

Consider the abstract Cauchy problem

2(0, ; ) : + = in 2(0, ; ), (0) =0

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where the separable Hilbert spaces , bounded and measurable operators() : , and 2(0, ; ), 0are given as above.

Proposition 4.6. [26, Proposition 2.3]

Assume the operators are uniformly coercive: there is a >0 such that

()()2, , [0, ]

Then there exists a unique solution of the Cauchy problem, and it satises

22(0;)(1)2(22(0;)+ 02)

This result can be extended. if and only if where () = (),

0 and is a solution of the proceeding Cauchy problem exactly when is thecorresponding solution of the problem

: + (() + ) = (), (0) =0

Corollary 4.1. A sufficient condition for existence by Proposition (4.6) is that there

exist a and >0 such that

()() +

2

2,

,

[0, ]

Similarly, uniqueness is obtained from such an estimate, even with= 0.

Theorem 4.3. If the coefficients of satisfy (A4) and (A5), then there exists a unique

weak solution of (4.25).

Proof. We choose = 2(), = 10 () for or = 1() for =,

then =1(). Lemma 4.4 implies

22()+ 2()10 ()

We obtain the desired result by Proposition 4.6.

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4.5 Uniqueness of fundamental solutions

Theorem 4.4. There exists a unique solution to (4.7).

Proof. Assume that there exist two fundamental solutions (, ) and (, ). Given

any initial condition 0 (), suppose the following parabolic equation with initialand boundary conditions

+ = in ,

= 0 on [0, ],

= on = 0,

has two solutions which can be expressed in terms of fundamental solutions (, ) and

(, )

(, ) =

( , )(),

(, ) =

( , )()

According to Theorem 4.3, the functions and are equal for a.e. [0, ]. Thus

(( , ) ( , ))() = 0 for a.e. (, ),

for every0 (). This implies

(, ) = (, ) for a.e. (, )

The partial integro-equation,

=

=1(, ) +

=1(, ) + (, )

(, )(, , )= 0

(4.36)

has a unique fundamental solution. Since the marginal density function(, ) of the

semimartingale () and the the marginal density function (, ) of the mimicking

process () both satisfy (4.36), the uniqueness of the fundamental solution of (4.36)

implies that () and () have the same marginal distributions.

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This is an extension of Theorem 3.3 for the case of a semimartingale. It is also

justied in [25, Theorem 3.4.2].

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Chapter 5

Markov Processes and Pseudo-Differential Operators

When we have a partial integro-differential equation, it is natural to investigate it using

pseudo differential operators. However, in the forward equation we derive, the integral

term is

(, )(, , ) which is not a standard convolution. So we cannotapply the general theory of pseudo-differential operators.

In this chapter, we review operator semigroups, Feller processes and discuss how

research of pseudo differential operators arises in the martingale problem. To end this

chapter, we indicate an area of further study. We plan to investigate the generator of

a semimartingale (). We rst want to show that is a pseudo-differential operator

with a symbol (,,). In principle we could check that (,,) satises certain

conditions in [6, Theorem 4.2] which should imply the existence of a Markov process

with generator . We believe that this could lead to a new proof of the mimicking

theorems, but this appears to be a challenging problem and we leave it for future

research.

5.1 Operator semigroups and Feller processes

In this section, we give a brief introduction to operator semigroups and their gener-

ators from a probabilistic perspective. We outline the relationship between operator

semigroups and Feller processes.

Denition 5.1. (Operator semigroups [18] Denition 4.1.1)

Let (, ) be a Banach space. Then a one parameter family of bounded linear

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operators()0 (, ) is called an operator semigroup if

+= , for all , 0, (5.1)

0=

We call()0 strongly continuous if

lim0

= 0,

for all.The semigroup ()0 is called a contraction semigroup, if for all0,

1

holds, that is, if each of the operators is a contraction. As usual,

denotes the

operator norm.

It is easy to see that (5.1) corresponds to the exponential Cauchy functional equation

( + ) =()(), (0) = 1,

where () is a nonnegative function from to . The solution to the exponentialCauchy functional equation is the family of exponential functions () = , .However, this family represents all possible solutions only if an additional assumption

of continuity is made. In fact, the assumption that () is continuous from the right in

the origin is already sufficient to make the functions () = the only solutions. We

now introduce a similar assumption for the operator semigroup dened by (5.1). Now

we want to show that by analogy to the Cauchy equation, an operator semigroup can

be represented in the form = for a suitable operator .

Denition 5.2. (Generators of semigroups [18] Denition 4.1.11)

Let()0 be a strongly continuous semigroup of operators on a Banach space(,

). The generator of()0 is dened by

()() = lim0+

() ()

, (5.2)

with domain

() =

lim0+

() ()

exists as strong limits.

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While the innitesimal generator is dened as the right-hand derivative of at

0, the derivative of at any point can be calculated by

()() = lim

0

((+ ))

,

and we have the following lemma.

Lemma 5.1. ([18] Lemma 4.1.14) Let ()0 be a strongly continuous semigroup of

operators on a Banach space (, ), and denote by its generator with domain(), then

(i) For any and 0, it follows that 0() and =

0

(ii) For() and 0, we have(), that is, () is invariant under, and

= =

(iii) For() and0, we get

=

0

=

0

The derivative is well dened on the domain () of and in fact the equation

= is Kolmogorovs backward equation and = is Kolmogorovs

forward equation.

From a stochastic point of view, operator semigroups start from the study of Markov

processes.

Denition 5.3. Given a Markov process, we can dene the corresponding family ofoperators() for0 by

()() =[(())() =], (5.3)

for each (), , where() denotes the space of bounded Borel mea-surable functions on.

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If a Markov process () is time-homogeneous, we can write =.

Theorem 5.1. Let () be a time-homogeneous Markov process, then the transition

operators()0 form a semigroup.

Proof. We want to show that +() =() holds for any(). We have

() =0() =[(())] =()

Then, because of the Markov property of, we get

(()) =0() =[(())] =[()[(())]]

=[[(+)]] =[(+)] =+()

Hence,+ = , ()0 form a semigroup.

Example 5.1. (The generator of a compound Poisson process)

Let() be a sequence of independent and identically-distributed random variables

with distribution function()and letbe a Poisson process with intensity. Denote

=1+ + . The compound Poisson process is dened by

() =1

1=

The transition operator of is given by

= [(())],

where we assume that(). To simplify calculations, we dene an operator by

() :=[( + 1)] =

( + )()

and note that

() :=[( + 1+ + )] =[( + )]

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Now it holds that

()() =[(())] =

0

[( + )]( = )

=0 ()

! [( + )]

=0

()

! ()

=

()

(),

and the transition semigroup can be written as where is given by

() =( )() =

( + ) ()

()

Example 5.2. The generator of a Levy process)

Let (())0 be a Levy process on with characteristic triple (,,). Then the

generator of is dened for any0()

()() =1

2

=1

2

() +

=1

()

=

(( + ) ()

=1

()11)()

Now we dene the Feller process, a type of process that is essentially a Markov

process satisfying some additional mild regularity assumptions.

Denition 5.4. (Feller process [18] Denition 4.1.4) Let()0 be a strongly contin-

uous semigroup of operators on a Banach space ((, ), ) which is positivepreserving, i.e. 0 yields0. Then()0 is called a Feller semigroup.

A Markov processwith transition semigroup ()0 is a Feller process if()0

is a Feller semigroup.

The class of Feller processes includes Levy processes, Dupire local volatility processes

and affine processes in nance. Feller processes may have nonstationary increments,

while Levy processes necessarily have stationary increments.

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Recall that the positive maximum principle [12, Thereom 4 in Section 6.4] also holds

for an elliptic second order differential operator

=

=1() +

=1()+ ()

The connection to semigroups is made by fact that generators of Feller processes satisfy

the same maximum principle.

Proposition 5.1. (Maximum principle [18, Theorem 4.5.1])

Let ()0 be a Feller semigroup on (, ) with generator (, ()), ()(, ). Then satises the positive maximum principle; that is, for()suchthat for some0 the fact that(0) = sup()0 implies that(0)0.

Proof. Suppose that () and that for some 0

we have (0) = sup()0. Since each of the operators ,0 is positivity preserving we nd that

()(0) = (+)(0) ()(0)(+)(0) += (0)

which implies

(0) = lim0

(0) (0)

0

The fact that generators of a Feller semigroup and elliptic operators satisfy thepositive maximum principle suggests a connection between them. Denote by (

)

the class of functions on which are innitely differentiable and have compact support.

Then we recall the following theorem.

Theorem 5.2. [19] Ifis a continuous Feller processes on[0, ]with operator and

()(), then is elliptic.

5.2 Pseudo-differential operators

In the preceding section we have seen that if is a continuous Feller process, its

generator is a second order elliptic differential operator. As an example, consider an

-valued Levy process and its transition operator:

() =[(())(0) =] =

( + )()

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The second equation is true because of independence and stationarity of increments;

is a probability measure. The Fourier transform of is

() =

() =[

()(0) =] =(),

where is the characteristic exponent of the Levy process . Using the convolution

theorem, we get () = () =()() =()()The inverse Fourier transform gives

() = 1

2

()()The generator ofis given by

() = lim0

() ()

= 1

2

() 1

()

= 1

2

()()In general, operators with such a representation are called pseudo-differential operators.

We give a formal denition, beginning with

Denition 5.5. (Continuous negative denite functions)

A function : is continuous negative denite if it is continuous and if, forany choice of and vectors1, , , the matrix

(() + () ( ))=1

is positive Hermitian, i.e. for all1, , ,

=1

(() + () ( ))=10

Some typical examples of continuous negative denite functions are:

for (0, 2],

1 for 0,

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log(1 + 2) + arctan .

We now recall

Denition 5.6. (Pseudo-differential operators) [18]

Let (, ()) be an operator with0 () (). Then is a pseudo-differential

operator if

()() =(, )() (5.4)

=(2)2

(, )(),

for0 ().() = (2)2 ()

is the Fourier transform of . The symbol (, ) : is locally boundedin(, ), (, ) is measurable for every, and(, ) is a continuous negative denitefunction for every.

Time-inhomogeneous processes

Denition 5.7. The family of generators of is dened by

= lim0+

, (5.5)

for all >0, (), where() denotes the domain of.

Denition 5.8. Let()>0 be a family of operators with0 ()(). Then

is a pseudo-differential operator for all >0 if

()() =(,,)() (5.6)

=(2)2

(,,)(),

for0 (), () = (2)2

(), is the Fourier transform of. The

symbol(,,) : + is locally bounded in(, ),(, , )is measurablefor every, , and(,, ) is a continuous negative denite function for every(, ).

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5.3 Construction of a Markov process using a symbol

Time-homogeneous case

In [17], Hoh showed that if a symbol of a pseudo-differential operator satises certain

conditions, then one can construct a unique Markov process whose generator has that

symbol.

Let (, ) be a separable metric spaceand let denote the space of all cadlag

paths with values in ,

:=: [0, ), is right continuous, lim

() exists for all >0,

let () denote the set of probability measures on .

Denition 5.9. [17] Let be a linear operator with domain (). A probability

measure () is called a solution of the martingale problem for the operatorif for every(), the process

(()) 0

(())

is a martingale under with respect to the ltration =F()0.If for every probability measure (), there is a unique solution of the

martingale problem for with initial distribution

(0)1 =,

then the martingale problem for A is calledwell posed.

We assume that : is a continuous negative denite reference function forsome >0, and >0. Dene () = (1 + ())12, .

Theorem 5.3. ([17])

Let: be a continuous negative denite symbol such that(, 0) = 0 forall . Letbe the smallest integer such that

>(

2) +

and suppose that

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(i) (, ) is (2+ 1 ) times continuous differentiable with respect to and forall , 2+ 1 ,

(, ) 2() (5.7)

holds for all , .

(ii) For some strictly positive function : +,

(, )() 2(), (5.8)

for all, 1. Then the martingale problem for the operator(, )with domain0 (

) is well-posed.

We illustrate theorem 5.3 with some examples. Let the continuous negative denite

reference function () be 2, then () = (1 + 2)12.

Examples:

1. Brownian motion(), the symbol of its generator is

(, ) =1

22

2. Geometric Brownian motion (), () = ()+()(), the symbol

of its generator is

(, ) =1

2222

3. CIR process (), () = (())()(), the symbol of itsgenerator is

(, ) =1

222 + ( )

4. Levy process() with Levy triplet (,,), the symbol of its generator is

(, ) =() + 12

22 +

(1 +

1 + 2)()

5. A continuous diffusion process(),() =(())() where(()) ia an

adapted process, the symbol of its generator is

(, ) =1

2()22

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In order to satisfy (5.7), when = 3, = 1,() needs to be 6 times continuous

differentiable with respect to. If() = 14()2, then (5.8) holds for all 1.

Time-inhomogeneous case

Bottcher showed that we could construct a time-inhomogeneous Markov process using

pseudo-differential operators.

Denition 5.10. ([6]) A continuous negative denite function: belongs tothe class if for all0 there exists constants0 such that

(1 + ()) (1 + ())2(2)2

Denition 5.11. ([6]) Let ,0, 1, 2and. A function : + is in the class if for all , 0 and for any compact + there areconstants0 such that

(,,) (1 + ())()2

holds for all , and. Here is called the order of the symbol.Furthermore, the notation( ) is used if, for,

(,,,) ( )(1 + ())()2

holds, where the constants are independent of and .

We now recall

Theorem 5.4. ([6, Theorem 4.2])

Suppose is a negative denite function, and there exist0 >0, 0 >0 such that()

0

0 for all large. If a pseudo-differential operator with symbol (,,)

satises the following conditions,

(, , ) is a continuous function for all , ,

(,, ) is continuous negative denite for all +, ,

(,, 0) = 0 holds for all and,